This is a (mostly expository) paper on Reidemeister classes, twisted
Burnside-Frobenius theory, congruences, R-infinity property and all that. It
was written in 2005 and published in 2008. We post it as it was, only the
bibliography data is updated. For some of the recent progress see e.g.
arXiv:0903.4533, arXiv:0903.3455, arXiv:0802.2937, arXiv:0712.2601,
arXiv:0704.3411, arXiv:math/0703744, arXiv:math/0606725, arXiv:math/0606764,
arXiv:0805.1371 and references there.